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Mathematical Achievements of Pre-Modern Indian Mathematics

T. K. Puttaswamy
Publication Date: 
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Elsevier Insights
[Reviewed by
Kim Plofker
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This substantial volume offers a wide selection of demonstrations and examples of the methods of pre-modern Indian mathematics, especially those dealing with indeterminate equations. Readers interested in the worldwide development of mathematics will find this book’s abundance of detailed source material, presented in modern notation, a welcome change from the scantiness of the chapters on India in most general histories of mathematics. However, they may find some of its presentation confusing or difficult.

The author’s preface rather misleadingly states that “[t]he present book is chronological in its plan, covering all the contributions in Algebra, Geometry, and Hindu Trigonometry of one ancient Indian mathematician to the next from Vedic Period (800 BC) to the seventeenth century of Christian era” (p. xiii). To treat all such contributions in a field as densely populated as Indian mathematics over a period spanning more than 2500 years is impossible even in nearly 750 pages.

The survey does touch on a commendable variety of underrepresented and largely unknown mathematical sources, including some post-Vedic commentators on the ancient Śulvasūtra ritual geometry texts, influential medieval Jaina mathematicians, and post-1500 commentators on Sanskrit mathematical classics such as Jñānarāja (fl. 1503) and Kṛṣṇa (late sixteenth century). But there is apparently no mention (the lack of an index makes it difficult to be sure) of many other interesting authors such as Lalla, one of whose trigonometric rules inspired the great Bhāskara II to put forth one of the few known examples in Sanskrit mathematics of explicitly geometric arguments adduced to disprove an accepted formula; Vaṭeśvara, a tenth-century astronomer also well known as an author on trigonometry; Nityānanda, an early seventeeth-century astronomer at the Mughal court in Delhi who presented Sanskritized versions of Greco-Islamic mathematical astronomy; and Munīśvara in the mid-seventeenth century, known both for his major commentary on the work of Bhāskara II and for his astronomical disputes with some of the mathematicians mentioned by Puttaswamy on pp. 690–691. An examination of David Pingree’s encyclopedic Census of the Exact Sciences in Sanskrit (Philadelphia, 1970–1994) or Jyotiḥśāstra (Wiesbaden, 1981), neither of which is cited in the work under review or listed in its bibliography, would have prompted a more realistic assessment of the scope of the subject and the impossibility of treating it in a truly comprehensive fashion in any book-length survey.

Searching in this volume for information about a particular Indian mathematician is further complicated by the hybrid of chronological and regional/cultural schemes employed to organize the subject matter. For instance, the chapter on “Kerala Astronomers” is placed between chapters on Nārāyaṇa Paṇḍita (fl. 1356) and “Sixteenth- and Seventeenth-Century Commentators on Bhāskara II” but contains authors known or thought to have worked in the Kerala region as early as the seventh century and as late as the early nineteenth. The pre-Bhāskara II mathematicians it names are not mentioned in earlier chapters.

Many authors in other sections of the book, such as medieval commentators on the ancient Śulvasūtras, are likewise presented out of chronological sequence. Consequently, it is not at all straightforward to trace “the contributions of one ancient Indian mathematician to the next” in many cases. A timeline listing the names and known or estimated dates of every mathematician mentioned would be a helpful addition to the book.

The sheer mass of mathematical content in the many worked examples and rationales forces a certain amount of skimping on the historical context. There is no explanation of why the chronological endpoint of the book’s coverage was chosen to be the seventeenth century (though as noted above, this limit is not strictly observed). Very little is said about the personal backgrounds or social environments of most of the mathematicians discussed (many of whom, it must be acknowledged, have left only a very sparse biographical record). Several are introduced briefly without any bibliographic citations to assist the reader seeking further information, such as Rāma (p. 29) and Śrīpati (pp. xi, 326), while some are merely named with no identifying details whatever, e.g., Kapardisvāmin, Karavindasvāmin, and Sundararāja (pp. 52, 69).

What historical information there is for the most part faithfully summarizes reliable research, but many of the author’s statements are questionable or would benefit from additional explanation. For example, to say that “Brahmagupta initiated a new branch of Mathematics, Interpolation theory” (p. 204) because he has given the first known rule for second-order interpolation implies much fuller knowledge than is available to us about the relation between Brahmagupta’s work and that of his (largely unknown) predecessors. While this is arguably just quibbling over details of how to interpret and assign priority in mathematical discoveries, other elisions and misstatements in the book may lead to more serious misunderstandings. Claiming, for instance, that the famous compendium of practical mathematics written on birch-bark and known as the Bakhshālī Manuscript is “estimated to belong to third to seventh century AD” (p. 85) ignores scholarship in, e.g., Takao Hayashi’s authoritative Bakhshālī Manuscript (Groningen, 1995) that places its most likely date somewhere between the eighth and twelfth centuries. Certainly, many of the arithmetic techniques it describes are at least several centuries older than the manuscript itself, but that is just the sort of distinction that most of the book’s very brief historical descriptions do not make adequately clear.

For instance, the identification of the sixth-century astronomer Varāhamihira as a “Magadha Brahmin” (p. 141) recounts without critically examining an ancient legend conflating the toponym of Magadha centered in northeast India with the social group known as Maga Brāhmaṇas primarily associated with western India, where Varāhamihira lived. The eighteenth-century Swiss mathematician Lambert is inaccurately characterized as the first scholar outside the Indian mathematical tradition “who enunciated that π is an irrational number” (p. xxi). Although Lambert was indeed the first to give a formal proof of that fact, non-Indian assertions of the impossibility of knowing the ratio of the diameter to the circumference of a circle date back at least to Maimonides in the twelfth century.

The statement that Muñjāla and Bhāskara II “conceived of the differential calculus” (pp. 529–530) is a very broad interpretation of certain specific remarks they made about the behavior of a few specific quantities of the type we now call trigonometric functions. Their observations about this behavior are important and interesting in their own right as ideas on mathematical relationships involving vanishingly small differences, but casually identifying them with “differential calculus” as a whole, without further qualification, is more confusing than informative. Where historical details do come in for closer scrutiny they are sometimes explained in a technical way very difficult for the non-Sanskritist to grasp, as in the assertion that the scribe of the Bakhshālī Manuscript would have referred to that text as “‘kritam’ or ‘vrichitam’ instead of ‘likhitam’” by himself (p. 85) if he were also its author. (This refers to a distinction between Sanskrit words meaning “made” or “said”, generally applied to authors, and ones meaning “written” which generally though not exclusively apply to scribes, but the reader who does not already know that will not be likely to figure it out from the explanation as given.)

The book’s chief flaws are editorial rather than authorial failings, but they are serious enough to cause significant problems for the non-specialist reader. The copyediting process should have cleaned up not only numerous minor errors in typography, grammar and style (e.g., “a lesser intelligent person”, p. 143) but also the incoherence of the citations. Published works are sometimes cited with explicit publication data rather than a reference number in the bibliography, and sometimes vice versa (see p. 30 for examples of both conventions). Bibliography reference numbers when given are not always accurate, as in the citation of Sachau’s translation of al-Bīrūnī’s India as “Ref. [27]” (p. 161); it is actually [107], where the translator’s name is spelled “Sachu”. Frequently a source text or a secondary work is mentioned with no citation of any kind (as in the quotation from Āmarāja and the accompanying paraphrase from S. B. Dikshit on p. 141).

In the bibliography itself, the valuable citations of a number of important and/or little-known works are obscured by many formatting problems. It is nominally ordered alphabetically by author’s surname, but the last fourteen (of 163) items are completely out of sequence. Several sources cited in the book are missing from the bibliography (e.g., Rodet — misspelled “Rodét” — on p. 30, Genucchi and Dickson on p. 479). Variant spellings of the same author’s name in the bibliography and/or the text (Datta/Dutta, Mazumdar/Mazumder for an author more commonly referred to as Majumdar) add to the confusion, as does the occasional substitution of an initial for a surname (e.g., “Agathe K.”, “Christopher M.”, “David M.” for Keller, Minkowski and Mumford respectively; “Sastri K” for T. S. Kuppanna Sastri).

The transliteration and translation conventions for quoted Sanskrit excerpts are similarly inconsistent. Sometimes an original source is quoted in roman transliteration and translated (e.g., pp. 594–598); sometimes it is quoted in nāgarī script and translated (e.g., p. 600); sometimes it is quoted in nāgarī and only paraphrased in English (e.g., p. 105, where the paraphrase inaccurately implies that the mathematician Āryabhaṭa explicitly claimed to have composed his Āryabhaṭīya in year 3600 of the Kaliyuga period, although the quoted verse actually states only that Āryabhaṭa was 23 years old in that year); and sometimes the nāgarī quote is not translated at all (e.g., p. 588, where the Kerala astronomer Nīlakaṇṭha is said to pay “rich tribute” to his predecessor Parameśvara in a quoted remark that remains unexplained).

To take just one of hundreds of examples of unsystematic transliteration, the famous treatise of Āryabhaṭa is called the “Aryabhateeya” on p. 105 and “Aryabhatiya” and “Aryabhatiam” on p. 143. Diacritical marks are applied inconsistently from one Sanskrit word to another, and even within the same transliterated name, as in the spelling “Ganeśa” for Gaṇeśa (p. 690). The nāgarī text also suffers from frequent typographical errors, such as “pūvaṃ” for “pūrvaṃ” and “tiryaḍ” for “tiryaṅ” (p. 18), “mauviṃ” for “maurviṃ” (p. 399), and so forth.

These caveats are intended to warn rather than deter non-Indologist readers turning to Puttaswamy’s book for its cornucopia of useful modern descriptions of various pre-modern Indian mathematical techniques. I note that the copyright page of this Elsevier Insights volume promises a (forthcoming?) online version, so it is to be hoped that the publisher and/or the author will have an early opportunity to improve the editing and add an index. Especially if the author is serious about intending the work as a possible textbook for undergraduate or graduate courses (p. xiii), its extensive information about Indian mathematics needs to be contextualized in a way that students (and their teachers) will find clear, unambiguous and accessible.

Kim Plofker is Assistant Professor of Mathematics at Union College in Schenectady, NY, and the author of